Proof of the Division Algorithm

[matqkks]

In many books on number theory they define the well ordering principle (WOP) as:
Every non- empty subset of positive integers has a least element.
Then they use this in the proof of the division algorithm by constructing non-negative integers and applying WOP to this construction. Is it possible to apply the WOP to a subset of non-negative integers? Am I being too pedantic?

 

[HallsofIvy]

It's rather obvious isn't it? "Applying the WOP to a subset of non-negative integers" would simply mean that, given X, a subset of the non-negative integers, any subset of X has a least member. And that is true because any subset of X is also a subset of the non-negative integers.

If that is not what you mean then please explain what you mean by "apply the WOP to a subset of non-negative integers".​

 

[matqkks]

Yes of course. I just had a senior moment.
Thanks.

 

[HallsofIvy]

There are those of use who live in "senior moments"! We are called "seniors".